3
Part of 2023 Mexican Girls' Contest
Problems(2)
Islands in Máxico
Source: 2nd National Women´s Contest of Mexican Mathematics Olympiad 2023 , level 1+2 p3
7/23/2023
In the country Máxico are two islands, the island "Mayor" and island "Menor". The island "Mayor" has states, with exactly cities each one. The island "Menor" has only one state with cities. "Aeropapantla" and "Aerocenzontle" are the airlines that offer flights in Máxico. "Aeropapantla" offer direct flights from every city in Máxico to any other city in Máxico. "Aerocenzontle" only offers direct flights from every city of the island "Mayor" to any other city of the island "Mayor".If the percentage of flights that connect two cities in the same state it´s the same for the flights of each airline, What is the least number of cities that can be in the island "Mayor"?
Mexico
equation on real numbers
Source: 2nd National Women's Contest of Mexican Mathematics Olympiad 2023 , problem 3 teams
7/23/2023
Find all triples of real numbers all different from zero that satisfies: \begin{eqnarray} a^4+b^2c^2=16a\nonumber \\ b^4+c^2a^2=16b \nonumber\\ c^4+a^2b^2=16c \nonumber \end{eqnarray}
equationalgebraMexico